This clinical trial studies how well intensity modulated proton therapy (IMPT) or intensity modulated X-ray (radiation) therapy (IMRT) works after surgery in treating patients with head and neck cancer. IMPT is a type of radiation therapy that allows for the most accurate application of proton radiation to the tumor and has the potential to reduce treatment-related side effects. IMRT is a type of 3-dimensional radiation therapy that uses computer-generated images to show the size and shape of the tumor. Thin beams of x-ray radiation of different intensities are aimed at the tumor from many angles. This type of radiation therapy reduces the damage to healthy tissue near the tumor. IMPT may work as well as IMRT after surgery in treating patients with head and neck cancer.

PRIMARY OBJECTIVES: I. To evaluate the local-regional control among subjects in each arm at 2 years after study registration. II. To evaluate difference in MD Anderson Dysphagia Inventory (MDADI) score between patients with proton and intensity-modulated radiation therapy (IMRT) adjuvant radiation. SECONDARY OBJECTIVES: I. To determine overall survival, progression free survival, local, regional, distant recurrence risks, and infield and outfield recurrence in the trial at 2 years after study registration. II. To determine the rate of grade 3+ acute adverse events (from treatment start to 30 days after radiation completion date). III. To determine the rate and severity of late solicited toxicities. IV. To determine the incidence of secondary acute effects attributable to radiotherapy (e.g., percutaneous endoscopic gastrostomy \[PEG\] tube placement, duration and dose of narcotic analgesia required, weight loss, and hospitalization days). V. To determine the impact of treatment on patient-reported quality of life. VI. To objectively quantify the severity of oral mucositis during and following radiotherapy. EXPLORATORY OBJECTIVES: I. To estimate direct and indirect costs of the study regimen and compare these with standard of care treatment techniques. II. To correlate histopathologic, molecular, and tumor genetic/epigenetic alterations with clinical outcomes. III. To correlate circulating biomarkers (micro ribonucleic acid \[miRNA\], circulating tumor deoxyribonucleic acid \[ctDNA\]) with clinical outcomes. IV. To determine adverse events and patient reported outcomes related to abbreviated concomitant chemotherapy. V. To qualitatively evaluate patient beliefs regarding tradeoffs of cancer control, treatment time, cost, acute side effects, and late side effects. VI. To determine cancer outcomes, adverse events, and patient reported outcomes and compare across head and neck subsites, between those aged ≥ 65 to those age \< 65 at date of enrollment, between male and female, and in the adjuvant population between time to total package completion (\< 9 weeks vs ≥ weeks, surgery = day 0), and by treatment with and without chemotherapy. VII. To evaluate the predictive relationship of linear energy transfer (LET) weighted modeling using an relative biologic enhancement (RBE)-based model and RBE-independent model with grade 3+ acute and late toxicity. OUTLINE: Patients who already underwent surgical resection are assigned to Arm A. Patients who have undergone upfront surgical resection and need postoperative radiotherapy are assigned to Arm B or Arm C. ARM A: Patients undergo intensity modulated proton therapy (IMPT) or IMRT for 18 sessions over 24 days in the absence of disease progression or unacceptable toxicity. Patients may receive cisplatin intravenously (IV) over 1-2 hours per standard of care. Patients undergo computed tomography (CT) or magnetic resonance imaging (MRI) and positron emission tomography (PET)/CT or PET/MRI during screening and follow-up. Patients may also undergo blood sample collection throughout the trial. ARM B: Patients undergo IMPT for 15 sessions over 19 days in the absence of disease progression or unacceptable toxicity. Patients may receive cisplatin IV over 1-2 hours per standard of care. Patients undergo CT or MRI and PET/CT or PET/MRI during screening and follow-up. Patients may also undergo blood sample collection throughout the trial. ARM C: Patients undergo IMRT for 15 sessions over 19 days in the absence of disease progression or unacceptable toxicity. Patients may receive cisplatin IV over 1-2 hours per standard of care. Patients undergo CT or MRI and PET/CT or PET/MRI during screening and follow-up. Patients may also undergo blood sample collection throughout the trial. After completion of study treatment, patients are followed up within 21 days and then at months 3, 6, 9, 12, 15, 18, 24, 35, 48, and 60.